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Mirrors > Home > MPE Home > Th. List > rexbid | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.) |
Ref | Expression |
---|---|
rexbid.1 | ⊢ Ⅎ𝑥𝜑 |
rexbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexbid | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexbid.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | rexbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rexbida 3277 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1785 ∈ wcel 2111 ∃wrex 3107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-rex 3112 |
This theorem is referenced by: rexbidvALT 3280 rexeqbid 3369 scott0 9299 infcvgaux1i 15204 bnj1463 32437 fvineqsneq 34829 poimirlem25 35082 poimirlem26 35083 elrnmptf 41807 smfsupmpt 43446 smfinfmpt 43450 |
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