Theorem bnj609 30241

Description: Technical lemma for bnj852 30245. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj609.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj609.2	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
bnj609.3	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj609	⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑋

Allowed substitution hints:   𝜑(𝑓)   𝐺(𝑓)   𝜑″(𝑓)

Proof of Theorem bnj609

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj609.2	. 2 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
2		bnj609.3	. . 3 ⊢ 𝐺 ∈ V
3		dfsbcq 3404	. . 3 ⊢ (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑 ↔ [𝐺 / 𝑓]𝜑))
4		fveq1 6102	. . . 4 ⊢ (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅))
5	4	eqeq1d 2612	. . 3 ⊢ (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
6		bnj609.1	. . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
7	6	sbcbii 3458	. . . 4 ⊢ ([𝑒 / 𝑓]𝜑 ↔ [𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8		vex 3176	. . . . 5 ⊢ 𝑒 ∈ V
9		fveq1 6102	. . . . . 6 ⊢ (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅))
10	9	eqeq1d 2612	. . . . 5 ⊢ (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)))
11	8, 10	sbcie 3437	. . . 4 ⊢ ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
12	7, 11	bitri 263	. . 3 ⊢ ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
13	2, 3, 5, 12	vtoclb 3236	. 2 ⊢ ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
14	1, 13	bitri 263	1 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ∅c0 3874  ‘cfv 5804   predc-bnj14 30007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by:  bnj600  30243  bnj908  30255  bnj934  30259