Theorem bj-drnf2v 31939

Description: Version of drnf2 2318 with a dv condition, which does not require ax-13 2234. Could be labeled "nfbidv". Note that the version of axc15 2291 with a dv condition is actually ax12v2 2036 (up to adding a superfluous antecedent). (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-drnf2v.1	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-drnf2v	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem bj-drnf2v

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦
2		bj-drnf2v.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	nfbidf 2079	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699

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-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)