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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 rule). (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 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
4 | 1, 3 | rexbida 3047 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 Ⅎwnf 1708 ∈ wcel 1990 ∃wrex 2913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-nf 1710 df-rex 2918 |
This theorem is referenced by: rexbidvALT 3053 rexeqbid 3151 scott0 8749 infcvgaux1i 14589 bnj1463 31123 poimirlem25 33434 poimirlem26 33435 elrnmptf 39367 smfsupmpt 41021 smfinfmpt 41025 |
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