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Mirrors > Home > MPE Home > Th. List > rexbidv2 | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.) |
Ref | Expression |
---|---|
rexbidv2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
Ref | Expression |
---|---|
rexbidv2 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexbidv2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) | |
2 | 1 | exbidv 1837 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))) |
3 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
4 | df-rex 2902 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)) | |
5 | 2, 3, 4 | 3bitr4g 302 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 |
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 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-rex 2902 |
This theorem is referenced by: rexbidva 3031 rexss 3632 exopxfr2 5188 isoini 6488 rexsupp 7200 omabs 7614 elfi2 8203 wemapsolem 8338 ltexpi 9603 rexuz 11614 lpigen 19077 llyi 21087 nllyi 21088 elpi1 22653 xrecex 28959 bnj18eq1 30251 ldual1dim 33471 pmapjat1 34157 mrefg2 36288 islssfg2 36659 fourierdlem71 39070 hoiqssbl 39515 |
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